Beamforming method for wireless communication systems and apparatus for performing the same

ABSTRACT

An improvement in a method of transmit beamforming between a transmitter ( 16 ) and a receiver ( 18 ) for a time varying lading channel comprises the step of performing transmit beamforming using less than complete knowledge of the previous fading blocks to design a codebook ( 26   a ) of a current fading block with each time frame. One embodiment comprises a successive beamforming algorithm and a second embodiment comprises a vector quantization beamforming algorithm. A fading parameter α is determined at least in the transmitter or receiver by monitoring a mobile Doppler frequency.

RELATED APPLICATIONS

The present application is related to U.S. Provisional PatentApplication Ser. No. 60/708,945, filed on Aug. 16, 2005, which isincorporated herein by reference and to which priority is claimedpursuant to 35 USC 119.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to methods wireless digital beamforming and inparticular to beamforming for a time varying fading channel.

2. Description of the Prior Art

Beamforming is a signal processing technique used with arrays oftransmitters or receivers that controls the directionality of, orsensitivity to, a radiation pattern. When receiving a signal,beamforming can increase the gain in the direction of wanted signals anddecrease the gain in the direction of interference and noise. Whentransmitting a signal, beamforming can increase the gain in thedirection the signal is to be sent. This is done by creating beams andnulls in the radiation pattern. Beamforming can also be thought of asspatial filtering.

Beamforming takes advantage of interference to change the directionalityof the array. When transmitting, a beamformer controls the amplitude andphase of the signal at each transmitter, in order to create a pattern ofconstructive and destructive interference in the wavefront. Whenreceiving, information from different sensors is combined in such a waythat the expected pattern of radiation is preferentially observed.

The signal from each antenna is amplified by a different “weight.” Whenthat weight is negative by just the right amount, noise received by thatantenna can exactly cancel out the same noise received by some otherantenna, causing a “null.” This is useful to ignore noise or jammers inone particular direction, while listening for events in otherdirections.

Conventional beamformers use a fixed set of weightings and time-delaysto combine the signals from the sensors in the array, primarily usingonly information about the location of the sensors in space and the wavedirections of interest. In contrast, beamforming techniques, generallycombine this information with properties of the signals actuallyreceived by the array, typically to improve rejection of unwantedsignals from other directions. As the name indicates, an beamformer isable to automatically adapt its response to different situations.

Transmit beamforming has been widely adopted for wireless systems withmultiple transmit antennas. For a block fading channel, the Grassmannianbeamformer has been shown to provide the best performance for givenamount of feedback. However, the original Grassmannian beamformer doesnot take the time domain correlation of the channel fading intoconsideration.

Multiple-input multiple-output (MIMO) systems offer much larger channelcapacity over traditional single-input single-output (SISO) systems.Recently, many transmit-beamforming algorithms have been developed toexploit the high capacity in the MIMO systems. The transmit-beamformingschemes require certain amount of channel state information (channelstate information) at the transmitter. Typically, the channel stateinformation is conveyed from the receiver to the transmitter through afeedback link. It has been shown in the art that, even with limitedfeedback, a good beamforming scheme can provide significant amount ofarray processing gain. In a slow fading environment, the performance ofthe transmit beamforming algorithms is usually better than that of theopen-loop algorithms (algorithms based on the space-time coding). Thisis because extra channel information is utilized to fine tune thetransmitted signal to fit the channel situation.

When perfect channel knowledge is available at the transmitter, theconventional eigen beamformer provides the best performance. However, ina practical wireless communication system, the feedback channel is bandlimited. The channel state information is quantized using only a fewbinary bits. It is of special interest to design efficient transmitbeamforming schemes that are based on finite rate feedback. In the art,a universal lower bound on the outage probability of finite feedbackbeamformer is established for a block fading model. It is demonstratedthat the relative loss in outage performance from finite feedback casedecreases exponentially with the number of feedback bits. In addition, adesign criterion has been introduced for finite rate beamforming. Thisdesign criterion minimizes the maximum inner product between any twobeamforming weights in the beamformer codebook, and the codebook problemis shown to be equivalent to the line packing problem in theGrassmannian manifold.

Several prior art beamformers have also been constructed based on thefinite rate feedback constrain. It has been shown that beamformingcodebook constructed based on the maximum receive SNR approach willresult in the same Grassmannian beamforming criterion as previouslyknown. Many beamforming codebooks have been constructed for thepractical MIMO systems. Recently, the Grassmannian beamforming algorithmis further investigated and its SNR performance as well as symbol errorrate have been accurately quantified for a given number of feedbackbits.

The above transmit beamforming algorithms assume a block fading model,i.e., the MIMO channels fade independently from one frame to the nextframe. However, in an ordinary wireless system, the actual channelcoefficients exhibit strong inter-frame correlation. An efficientbeamforming scheme should utilize this correlation and reduce the amountof feedback. A class of beamforming schemes that exploits theinter-frame correlation in channel fading have been previouslyintroduced. This class of algorithms is called gradient feedback (GFB)beamforming algorithm. The gradient feedback algorithm applies severalrandom perturbed transmission weights on top of the normal beamformingweight. Then a few feedback bits from the receiver 18 select theperturbed weight vector which provides the highest receive power. It hasbeen shown that the weight adaptation in gradient feedback can beapproximated by a coarse gradient adaptation. Furthermore, theperformance of the gradient feedback scheme has been analyzed in termsof convergence and tracking of an auto-regressive dynamic fading model(AR1).

BRIEF SUMMARY OF THE INVENTION

The illustrated embodiments of the invention are directed to transmitbeamforming for a time varying fading channel instead of to aquasi-static fading channel as in the prior art. The effect of Dopplershifting is considered in the illustrated embodiments. As a result, timedomain mutual correlations have been exploited in the channel fading.The resulting beamforming scheme works better in real worldtime-selective wireless systems than the prior art approaches. Theapproach of the invention is successive while the prior art methods arenot successive.

The illustrated embodiments of the invention relate to the applicationof vector quantization (vector quantization) technology on transmitbeamforming systems. In particular, the illustrated embodiments applyvector quantization technology With memories (including predictivevector quantization and finite state vector quantization) on transmitbeamforming systems. As a result, we successfully construct thepredicted vector quantization beamformer. The prior art disadvantageaddressed in this research is the design and evaluation of the specialbeamforming distance metric for the training of the generalized Lloydalgorithm.

Thus, the illustrated embodiments of the invention include a successivebeamforming algorithm for transmit beamforming systems using asuccessive beamforming adaptive codebook. As a result, we successfullyconstruct a time-varying beamforming adaptive codebook based on theknowledge from previous fading block. The major disadvantage addressedin this research is a systematic successive adaptive codebook strategythat can provide easy storage, synchronized adaptation, as well asdecent beamforming gains. Overall, the successive beamforming system canbe easily implemented.

The purpose of the invention is to improve the reliability, flexibility,data rate, and performance (in terms of SNR, bit error, or capacity) ofthe wireless systems.

One of the principles used in the illustrated embodiments is theexploitation of the time domain correlations in the channel fading toperform the transmit beamforming. Basically, the transmit weight fromthe previous fading block is used to carry out transmit beamforming onthe current fading block.

One advantage of the illustrated embodiments is that we use the samenumber of feedback bits for our transmit beamforming systems. However,we accomplish much better performance in terms of reliability, SNR,capacity, bit error rate than any existing prior art system. Theperformance gain is especially large at slow fading speeds.

Another major advantage is that the illustrated embodiments is that theyenjoy easy implementation. In the successive beamforming scheme, only asingle codebook is required on both sides of the wireless link. There isno need to have multiple codebooks for the different fading speeds.

Finally, our design approach is general and systematic. It can beextended to any number of transmit antennas, any number of feedbackbits, and any fading speeds.

The illustrated embodiments can be used for the existing and nextgeneration wireless communication systems. It can be adopted for anywireless system with multiple transmit antennas. For example, it be canadopted for 3G wireless system, or the residential WiFi or WiMaxsystems. Another major area of application is the communication systemsin the defense industry.

While the apparatus and method has or will be described for the sake ofgrammatical fluidity with functional explanations, it is to be expresslyunderstood that the claims, unless expressly formulated under 35 USC112, are not to be construed as necessarily limited in any way by theconstruction of “means” or “steps” limitations, but are to be accordedthe full scope of the meaning and equivalents of the definition providedby the claims under the judicial doctrine of equivalents, and in thecase where the claims are expressly formulated under 35 USC 112 are tobe accorded full statutory equivalents under 35 USC 112. The inventioncan be better visualized by turning now to the following drawingswherein like elements are referenced by like numerals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a simplified block diagram of a wireless system in which themethods of the invention are implemented.

FIG. 2 a is a functional block diagram illustrating a first embodimentusing a open-loop predictor.

FIG. 2 b is a functional block diagram illustrating a closed-loopencoder of the invention.

FIG. 2 c is a functional block diagram illustrating a closed-loopdecoder of the invention.

FIG. 3 is a graph comparing the signal-to-noise ratios for a pluralityof beamforming embodiments as a function of mobile speed for a timevarying fading channel in a numerical simulation of the invention wherethe number of transmit antennas N_(t)=3 and the number of feedback bits,N=3.

FIG. 4 is a graph comparing the signal-to-noise ratios for a pluralityof beamforming embodiments as a function of mobile speed for a timevarying fading channel in a numerical simulation of the invention wherethe number of transmit antennas N_(t)=4 and the number of feedback bits,N=4.

FIG. 5 is a graph comparing the bit error ratios (BER) for a pluralityof beamforming embodiments and mobile speeds as a function of SNR for atime varying fading channel in a numerical simulation of the inventionwhere the number of transmit antennas N_(t)=3 and the number of feedbackbits, N=3.

FIG. 6 is a graph comparing the bit error ratios (BER) for a pluralityof beamforming embodiments and mobile speeds as a function of SNR for atime varying fading channel in a numerical simulation of the inventionwhere the number of transmit antennas N_(t)=4 and the number of feedbackbits, N=4.

FIG. 7 is a graph comparing the bit error ratios (BER) for a pluralityof mobile speeds as a function of the parameters E_(s)/σ² and 1−η for atime varying fading channel in a numerical simulation of the inventionwhere the number of transmit antennas N_(t)=3 and the number of feedbackbits, N=3.

FIG. 8 is a graph comparing the bit error ratios (BER) for a pluralityof mobile speeds as a function of the parameters E_(s)/σ² and 1−η for atime varying fading channel in a numerical simulation of the inventionwhere the number of transmit antennas N_(t)=4 and the number of feedbackbits, N=4.

FIG. 9 is a graph comparing the bit error ratios (BER) of the enhancedadaptive SBF scheme for an AR1 channel for a plurality of mobile speedsas a function of SNR for a time varying fading channel in a numericalsimulation of the invention where the number of transmit antennasN_(t)=3 and the number of feedback bits, N=3. The value of the parameterη is determined using Table II.

FIG. 10 is a graph comparing the bit error ratios (BER) of the enhancedadaptive SBF scheme for an AR1 channel for a plurality of mobile speedsas a function of SNR for a time varying fading channel in a numericalsimulation of the invention where the number of transmit antennasN_(t)=4 and the number of feedback bits, N=4. The value of the parameterη is determined using Table II.

FIG. 11 is a graph comparing the bit error ratios (BER) of a GFBalgorithm and SBF algorithm of the invention for an AR1 channel for aplurality of mobile speeds as a function of SNR for a time varyingfading channel in a numerical simulation of the invention where thenumber of transmit antennas N_(t)=4 and the number of feedback bits, N=4and η=0:8702.

The invention and its various embodiments can now be better understoodby turning to the following detailed description of the preferredembodiments which are presented as illustrated examples of the inventiondefined in the claims. It is expressly understood that the invention asdefined by the claims may be broader than the illustrated embodimentsdescribed below.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Based on a first order auto-regressive (AR1) dynamic fading model, wedevelop two new classes of beamforming algorithms that exploit theinter-frame correlations in the channel fading. We first introduce analgorithm based on a predictive vector quantization (predictive vectorquantization) approach. The resulting predictive vector quantizationbeamformer accomplishes superior power delivery at the receiver 18.However, the error performance of the predictive vector quantizationbeamformer is not satisfactory at high SNRs, and it also has a highimplementation complexity.

To resolve these issues, we then develop a novel successive beamforming(SBF) algorithm. The new SBF scheme uses the knowledge of the previousfading blocks to aid the beamforming adaptive codebook of the currentfading block. The optimal codebook is constructed based on thesuccessive partition of the surface of a spherical cap. The new SBFscheme accomplishes superior performance at both low and high SNRs.Furthermore, it also enjoys easy implementation. Through numericalsimulations, we demonstrate that the disclosed predictive vectorquantization beamformer and successive beamformer outperform severalother previously known beamformers at various fading scenarios.

For the sake of simplicity the illustrated embodiments focus on thedesign and evaluation of good beamforming schemes for the multiple inputsingle-output (MISO) wireless systems. However, the results can beeasily extended to a MIMO system. To model the time varying nature ofthe channel fading, we adopt an AR1 fading model from B. Banister and J.Zeidler, “A simple gradient sign algorithm for transmit antenna weightadaptation with feedback,”

IEEE

Trans. Signal Processing, vol. 51, pp. 1156-1171, May 2003. Besides thefading model, our approach is novel from the prior art in that we do notuse a stochastic adaptation approach.

To exploit the inter-frame correlation in channel fading, we havedeveloped two classes of beamforming algorithms. The first one is basedon predictive vector quantization (predictive vector quantization). Theresulting predictive vector quantization beamformer accomplishessuperior power delivery as well as good error performance at lowsignal-to-noise ratios (SNRs). However, the error performance of thepredictive vector quantization beamformer is not satisfactory at highSNRs, and it also has a high implementation complexity.

To resolve these issues, we then develop a successive beamforming (SBF)algorithm, which is in essence a modified Grassmannian beamformingalgorithm. We evaluate the SNR performance for the SBF algorithm in astatic fading environment. We then analyze its tracking performance inthe AR1 fading environment using a simplified two-step approach. Basedon the steady state analysis, we calculate the optimal size of thesurface area of the spherical cap for adaptive codebook. To simplify theimplementation, we introduce a systematical adaptive codebook strategybased on a slow fading approximation. Finally, we use numericalsimulations to confirm that this new beamforming algorithm outperformsseveral existing beamforming algorithms.

A detailed system model and a SNR design target is disclosed. Inaddition, we present a novel predictive vector quantization beamformingscheme that accomplishes maximum receive power delivery. We introduce adesign criterion for the SBF systems. A systematic codebook constructionscheme is also illustrated. Finally, Monte Carlo simulations for theproposed beamforming algorithms are presented

In this specification bold uppercase (lowercase) letters denote matrices(column vectors); ()*; ()^(T); ()^(H) denote conjugate, transpose,and Hermitian operators, respectively; x_(ij) denotes the element at theith row and jth column of X; // denotes vector norm; U() denotesuniform distribution; Re(x); Im(x) stand for the real and imagine partof a complex variable x, respectively; p(); E(); cov() stand for theprobability density function (pdf), mean and covariance of a randomvariable respectively; 0 denotes the empty set; e₁=[1 0 . . . 0]^(T)denotes a column vector with every entries being zero except the firstentry. J₀() denotes the zeroth order Bessel function of the first kind;δ_(ij) denotes the discrete Dirac delta function.

It is to be further expressly understood that while the invention isdescribed in functional or mathematical terms as an efficient form ofexpression, the physical means for performing the various steps,processes, and functions disclosed below include any and all knownsoftware controlled digital and signal processors, computers, programmedlogic arrays, analog logic circuits and all electronic signal processingand conditioning circuits now known or later devised for performingequivalent functions.

Transmit Beamforming for Time-Varying Fading Channels System Model andPerformance Criterion

Consider a wireless system 10 with N_(t) transmit antennas 12 and asingle receive antenna 14 as shown in the system block diagram ofFIG. 1. Without loss of generality, we assume that the data istransmitted in frames. The digital input bits are provided to amodulator 22 in transmitter 16 which produces a parallel digital outputsignal s_(t) for the time frame. The parallel digital output signalss_(t) are signal condition for transmission as a single data stream bybase band circuit 23 and then multiplied or weighted by a weightingvector w_(t) by transmit multiplier 24. The weighting is taken from acodebook 26 which takes as its inputs a feedback channel 20 fromreceiver 18 and a delay circuit 22. The conditioned and weighted signalsare then provided to a plurality of antennas 12 for wirelesstransmission as vector channel signal h_(t). Codebook 26 a, delaycircuit 22 a and feedback channel 20 comprise in one embodiment theclosed loop decoder 36 of FIG. 2 c. As described below, FIGS. 2 a, 2 band 2 c describe a predictive vector quantization embodiment for theencoder comprised of codebook 26, delay circuit 22 and feedback channel20. Other implementations are possible, such as the successive feedbackembodiment also described below. These two embodiments, of course, donot exhaust the number or type of encoders 34 or decoders 36 which couldbe employed in FIG. 1 which operate consistent with the scope of theclaimed invention.

Thus, although data is transmitted from transmitter 16 to receiver 18 ash_(t), a second wireless transmission of a signal, i_(t), from receiver18 to transmitter 16 occurs by which an identical adaptive codebook 26 aand 26 b are established in data receiver 18 and data transmitter 16.The second transmission may be through the same set of antennas 12 and14, or through a separate or overlapping set of different antennas (notshown). Thus, it will be disclosed below, that the invention is animprovement in a method of transmit beamforming between a transmitterand a receiver for a time varying fading channel comprising the step ofperforming transmit beamforming using less than complete knowledge ofthe previous fading blocks by beamforming a codebook of a current fadingblock with each time frame. Two embodiments will be illustrated, namelya successive beamforming algorithm and a vector quantization beamformingalgorithm. In both embodiments a fading parameter a is determined atleast the transmitter or receiver by monitoring a mobile Dopplerfrequency. A best estimated channel direction {tilde over (g)}_(t) isgenerated based on the past channel inputs, namely {tilde over(g)}=S(g_(t−1)) where S() denotes a predicting function, {tilde over(g)}_(t) denotes a best estimate of the current channel direction basedon the previous channel direction g_(t−1).

The channel signal h_(t) is received at antenna 14 as a vector receivesignal r_(t). A channel estimator 28 evaluates the channel stateinformation and generates a feedback signal, i_(t), on the feedbackchannel 20 comprised of N bits/frame as described below which is inputto the beamforming codebook 26 in transmitter 16. It must be made clearthat the information which is fed back through the wireless channel 20is less than the complete channel information, and can be as little as asingle bit. The output of channel estimator 28 is coupled to a maximumlikelihood (ML) decoder 30 which reconstructs the transmitted signal asdisclosed below. The same form of codebook 26 is used in both thetransmitter 16 and receiver 18. Codebook 26 b, delay circuit 22 b andfeedback channel 20 comprise in one embodiment the closed loop encoder34 of FIG. 2 b. Again other implementations are possible, such as thesuccessive feedback embodiment also described below and the twodisclosed embodiments do not exhaust the different kinds of embodimentswhich could be employed in FIG. 1 which operate within the scope of theinvention.

At the i th frame, the receive signal in the MISO system is given by

r _(t) =h _(t) ^(H) w _(t) s _(t) +n _(t),  (1)

where the transmitted signal at time t is denoted as s_(t), the receivesignal as r_(t). In this specification as an example only, the signals_(t) represents a scalar value drawn from a phase-shift keying (PSK) orquadrature amplitude (QAM) constellation. Furthermore, the energy ofs_(t) satisfies E_(s)=E(/s_(t)/²). The noise parameter n_(t) is a randomvariable drawn from an independent identically distributed, zero-mean,white complex Gaussian process with cov(n_(t))=σ². The parameterh_(t)=(h_(1t); . . . ; h_(Ntt))^(T) represents the channel vector andthe entry h_(it); i=1; . . . ; N_(t) represents the channel path gain atthe i th transmit antenna at the t th frame. In this specification, thechannel vector h_(t) is assumed to be perfectly known at the receiver 18and it is generated according to an AR1 fading model

h _(t) =αh _(t−1) + αx _(t),  (2)

where x_(t) is an N_(t)×1 vector with each entry drawn from anindependent identically distributed zero-mean complex white Gaussianprocess, and its auto covariance function satisfies E(x_(t)x_(t)^(H))=I_(Nt). As a direct result of the independent identicallydistributed entries in x_(t), the channel path gains between any twotransmit antennas 12 are also uncorrelated, i.e.E(h_(it)h*_(jt))=δ_(ij). The parameter α in equation (2) defines thefading speed and its value is given in this embodiment by R. Clarke, “Astatistical theory of mobile radio reception,” Bell Syst. Tech. J., vol.47, pp. 957-1000, July 1968:

$\begin{matrix}{{\alpha = {J_{0}\left( \frac{2\pi \; {Tv}_{mb}}{\lambda_{t}} \right)}},} & (3)\end{matrix}$

where T denotes the duration of each data frame, and it also representsthe interval between two consecutive channel feedbacks. The parametersv_(mb) and λ, denote the mobile speed and carrier wavelength,respectively. In addition, the parameter α in equation (2) is simplydefined as α=√(1−α²). Note that a in equation (3) is a criticalparameter for a transmit beamforming system 10 since it describes thefading speed. A smaller α corresponds to faster fading speed. At theextreme case, v_(mb)→∞, then α→0, and the AR1 fading model falls back toa block fading model.

Typically, the fading parameter α is known at both the transmitter 16and receiver 18 since both sides can monitor the mobile Dopplerfrequency. Therefore, in the additional embodiments of the invention, weassume that the value of α is known at the transmitter 16. In apractical wireless system, it is also possible to have the receiver 18send the parameter α to the transmitter 16 for better performance. Sincethe fading speed changes relatively slower compared to the normal frameduration, the parameter α only needs to be sent back to the transmitter16 in some rare occasions, hence it incurs little implementationoverhead. The quantization on the fading parameter α will beinvestigated in greater detail below. We will demonstrate that arelative coarse quantization on α is sufficient for the proposedtransmit beamforming systems.

In equation (1), the parameter w_(t)

[w_(1t), . . . , w_(N) _(t) _(t)]^(T) represents the transmit weightvector at time t. The value of w_(t) is selected from a codebook usingan instantaneous signal-to-noise ratio criterion at the receiver 18

$\begin{matrix}{{w_{t} = {\arg \; {\max\limits_{c_{i} \in C}{{h_{t}^{H}c_{i}}}}}},} & (4)\end{matrix}$

where C={c₁, . . . , c₂ _(N) } represents the beamforming codebook. Thecodebook consists of 2^(N) beamforming weights. Each beamforming weightis an N_(t)×1 vector with unit norm, i.e., ∥c_(i)∥=1, ∀i. In FIG. 1,there is a feedback link 20 dedicated on conveying channel stateinformation from the transmitter 16 to the receiver 18. During eachframe of the data transition, the optimal beamforming weight is obtainedbased on equation (4) and its index is sent back to the transmitter 16using N binary bits.

The main goal is to provide a good beamforming scheme that can exploitthe channel correlation for the systems 10 in FIG. 1. Towards this end,we use the average receive signal-to-noise ratio as the performancemetric for the design and analysis of the beamforming system:

$\begin{matrix}{{{S\; N\; R} = {{\frac{E_{s}}{\sigma^{2}}{E\left( {{h_{t}^{H}w_{t}}}^{2} \right)}} = {\frac{E_{s}}{\sigma^{2}}{E\left( {\rho_{t}{{g_{t}^{H}w_{t}}}^{2}} \right)}}}},{{{where}\mspace{14mu} \rho_{t}}\overset{\Delta}{=}{{{h_{t}}^{2}\mspace{14mu} {and}\mspace{14mu} g_{t}}\overset{\Delta}{=}\frac{h_{t}}{h_{t}}}}} & (5)\end{matrix}$

represent the amplitude and the direction of the current channel,respectively.

-   -   a. For a block fading environment, it can be shown that g_(t)        and ρ_(t) are independent random variables. In addition,

$\begin{matrix}{{{p\left( \rho_{t} \right)} = \frac{\rho_{t}^{N_{t} - 1}^{- \rho_{t}}}{\left( {N_{t} - 1} \right)!}},} & (6)\end{matrix}$

and g_(t) is uniformly distributed on the surface of the unithypersphere, i.e., g_(t)˜U(Ω^(N) ^(t) ), where Ω^(N) ^(t) denotes thesurface of the complex N_(t)-dimensional unit hypersphere.

For the independent block fading case, the problem of optimalbeamforming adaptive codebook has been solved in the prior art. Theresulting design criterion is called the Grassmannian beamformingcriterion and it is summarized as follows as Lemma 1. For a MISO systemoperating in the independent block fading environment with N_(t)transmit antennas and N feedback bits, an ideal size-2^(N) beamformingcodebook is obtained by solving the following problem:

$\begin{matrix}{{C_{opt} = {\arg \; {\max\limits_{\forall C}{\min\limits_{\forall{i \neq \; j}}{d_{c}\left( {c_{i},c_{j}} \right)}}}}},} & (7)\end{matrix}$

where d_(c)(c_(i), c_(j))

√{square root over (1−|c_(i) ^(H)c_(j)|²)} denotes the chordal distancebetween two unit norm column vectors c_(i) and c_(j). In addition, thesignal-to-noise ratio performance of the ideal beamformer is upperbounded by

${S\; N\; R} \leq {\frac{E_{s}}{\sigma^{2}}{\left( {N_{t\;} - {\left( {N_{t} - 1} \right)2^{\frac{- N}{N_{t} - 1}}}} \right).}}$

The above Grassmannian beamformer is only optimal for the block fadingscenarios. For the time varying fading scenarios, the above designcriterion fails to exploit the inter-frame correlation betweenconsecutive frames. In what follows, we investigate a new adaptivecodebook strategy. The new design strategy will generate transmitweights based on the knowledge of the previous fading block. As aresult, the new scheme accomplishes much higher receive signal-to-noiseratio.

Beamforming Based on Predictive Vector Quantization

The beamforming problem above can be cast into an equivalent vectorquantization problem:

$\begin{matrix}\left\{ \begin{matrix}{{{{Source}\mspace{14mu} {input}\text{:}\mspace{14mu} h_{t}} = {{\alpha \; h_{t - 1}} + {\overset{\_}{\alpha}\; x_{t}}}};} \\{{{{{Codebook}\text{:}\mspace{14mu} C} = \left\{ {c_{1},\ldots \mspace{14mu},c_{N_{t}}} \right\}},{{{c_{i}} = 1};}}\;} \\{{{Distortion}\mspace{20mu} {metric}\text{:}\mspace{14mu} {\overset{\_}{d}\left( {h_{t},w_{t}} \right)}} = {{- {{h_{t}^{H}w_{t}}}^{2}} = {{- {{g_{t}^{H}w_{t}}}^{2}}{\rho_{t}.}}}}\end{matrix} \right. & (8)\end{matrix}$

A key observation is that the vector quantization problem in equation(8) and the vector quantization problem in P. Xia and G. Giannakis,“Design and analysis of transmit beamforming based on limited-ratefeedback,” IEEE Trans. Signal Processing, (to be published, hereinafter“Xia”) are quite different. The time-varying fading model in equation(2) can be considered a source with memory, whereas the block fadingmodel in Xia is simply a memory-less source. To construct a goodcodebook for the problem in equation (8), the standard memory-lessvector quantization with finite signal dimensionality becomes inferiorto the vector quantization algorithms with memory. An algorithm that canexploit time domain correlations should be able to provide betterperformance.

An effective strategy to incorporate memory into vector quantization isa predictive vector quantization technique. There are other vectorquantization schemes that exploit the memory from the signal source,e.g. finite state vector quantization (FSVQ) and combined FSVQ withtrellis encoding. However, these techniques have much higher complexityor encoding delays and not presently preferred. In what follows, we usea predictive vector quantization approach to solve the transmitbeamforming problem in equation (8).

The detailed structure of the predictive vector quantizer is depicted inFIGS. 2 a-2 c. There are three important blocks in FIGS. 2 a-2 c thatare not seen in the memory-less vector quantization: the linearpredictor block 22, the residue generator unit 24, and thereconstruction unit 26. The purpose of the predictor block 22 is toprovide the best estimated channel direction g, based on the pastchannel inputs.

In this specification, we adopt a simple open-loop approach to designthe optimal vector predictor 32. To clarify our derivation, a simpleillustration of the open-loop predictor 32 is depicted in the blockdiagram of FIG. 2 a. The open-loop approach has an attractive propertythat the predictor can be designed first without taking the quantizerinto consideration. In the open-loop scheme, the optimal estimation ofthe current channel direction is based on the actual channel direction,instead of the reconstructed channel direction, of the previous frame.The output from the predictor is expressed as:

{tilde over (g)} _(t) =S(g _(t−1)),  (9)

where S() denotes the predicting function, {tilde over (g)}_(t) denotesthe best estimate of the current channel direction based on the previouschannel direction g_(t−1). As shown in the schematic of FIG. 2 a thechannel direction {tilde over (g)}_(t) is input into a delay unit 38,which after one frame becomes the previous channel direction g_(t−1).Linear vector predictor 22 then generates the best estimate {tilde over(g)}_(t) of the current channel direction according to equation 9.

Once the predictor 32 has been designed, the residue generator block 24will convert the original channel vector h_(t) into a residue vector{tilde over (e)}_(t) as depicted in the schematic of FIG. 2 b in theclosed-loop encoder 34. Due to the special distance metric in equation(8), the simple subtractor unit in the prior art is not usable here. Inthe predictive vector quantization algorithms, the incorporation of theresidue generator unit 24 should not alter the original design target ofminimizing the overall quantization distortion. Towards this end, wedefine a simple linear residue generator 24.

We first introduce a vector space adaptation model based on complexHouseholder transform. Let H_(ouse)(y) denotes the complex Householdermatrix

${{H_{ouse}(y)} = {I - \frac{{uu}^{H}}{y^{H}u}}},$

where y is any column vector with unit norm and u

y−e₁. Using this Householder matrix, the actual channel direction g_(t)can be expressed as a linear function of the predicted channel direction{tilde over (g)}_(t) and its orthogonal complements:

g _(t) =H _(ouse)({tilde over (g)} _(t))v _(t)  (10)

where v_(t) is an N_(t)×1 column vector with unit norm ∥ν_(t)=1. Notethat the Householder matrix H_(ouse)( g _(t)) is a unitary matrix and ithas the property that the first column being g _(t) and the rest of thecolumns are orthogonal to {tilde over (g)}_(t). The physical implicationof equation (10) performed by residue generator 24 is that the actualchannel direction for the current frame is a linear combination of thepredicted channel direction and the subspace orthogonal to it, and thevector v_(t) contains the linear combination coefficients.

From equation (10), we define the residue vector as:

{tilde over (e)} _(t)

H_(ouse)({tilde over (g)} _(t))^(H) h _(t) =H _(ouse)({tilde over (g)}_(t))^(H) g _(t) ∥h _(t) ∥=v _(t) ∥h _(t)∥

The residue vector {tilde over (e)}_(t) is quantized inside the vectorquantization block 28 in FIG. 2 b, the quantized residue vector isdenoted as {tilde over (v)}_(t) and is obtained based on the instantminimum distortion criterion using the distance metric in equation (8):

$\begin{matrix}{{\hat{v}}_{t} = {{\arg \; {\min\limits_{c_{i} \in C}{\overset{\_}{d}\left( {{\overset{\sim}{e}}_{t},c_{i}} \right)}}} = {{\arg \; {\min\limits_{c_{i} \in C}{- {{{\overset{\sim}{e}}_{t}^{H}c_{i}}}^{2}}}} = {\arg \; {\min\limits_{c_{i} \in C}{{- {{v_{t}^{H}c_{i}}}^{2}}{{h_{t}}^{2}.}}}}}}} & (11)\end{matrix}$

The output from the vector quantization block 28, which is thetransmitted feedback signal, it, is used by the reconstruction units 26at both the encoder 34 in FIG. 2 b and the decoder 36 in FIG. 2 c, wherethe channel direction is reconstructed as:

ĝ _(t) =H _(ouse)({tilde over (g)} _(t)){circumflex over (v)}_(t).  (12)

Based on the reconstructed channel direction ĝ_(t), the transmit weightat the transmitter 16 is simply given by w_(t)=ĝ_(t). The transmittedfeedback signal it is inversely quantized by inverse VQ block 30 in thedecoder 36 to generate {circumflex over (v)}_(t).

The incorporation of the above residue generation unit 24 andreconstruction unit 26 does not change the original design target inequation (8), i.e., the quantization on the residue signal {tilde over(e)}_(t) is equivalent to the quantization on h_(t). To see thisequivalence, we rewrite the original design target (minimization ofaverage distortion) as:

$\begin{matrix}{\quad\begin{matrix}{{\min \; {E\left( {\overset{\sim}{d}\left( {w_{t},h_{t}} \right)} \right)}} = {\min - {E\left( {{{{\hat{g}}_{t}^{H}g_{t}}}^{2}{h_{t}}^{2}} \right)}}} \\{= {\min - {E\left( {{{{\hat{v}}_{t}^{H}{H_{ouse}\left( {\hat{g}}_{t} \right)}^{H}{H_{ouse}\left( {\overset{\_}{g}}_{t} \right)}^{H}v_{t}}}^{2}{h_{t}}^{2}} \right)}}} \\{= {\min - {E\left( {{{{\hat{v}}_{t}^{H}v_{t}}}^{2}{h_{t}}^{2}} \right)}}} \\{{= {\min \; {E\left( {\overset{\_}{d}\left( {{\hat{v}}_{t},{\overset{\_}{e}}_{t}} \right)} \right)}}},}\end{matrix}} & (13)\end{matrix}$

where the first equality is based on the fact w_(t)=ĝ_(t), the next twoequalities are due to equations (10) and (12) and the fact thathouseholder matrix H_(ouse)({tilde over (g)}_(t)) is unitary, and thelast equality is simply due to the definition of the distance metric inequation (8). Based on equation (13), the original minimum distortiondesign target is accomplished by the quantization on the residue signal{tilde over (e)}_(t).

The performance of a predictive vector quantization system is evaluatedby its output signal-to-noise ratio. For the above open-loop predictivevector quantization system, the output signal-to-noise ratio can beexpressed as the product of two terms

$\begin{matrix}{{S\; N\; R} = {\frac{E_{s}}{\sigma^{2}}S\; N\; R_{predictor}G_{{VQ},}}} & (14)\end{matrix}$

where SNR_(predictor)

E(|h_(t) ^(H){tilde over (g)}_(t|) ²)=E(|g_(t) ^(H){tilde over(g)}_(t)|²∥h_(t)∥²) denotes the output signal-to-noise ratio at thepredictor unit 32,

$\begin{matrix}{{G_{VQ}\overset{\Delta}{=}\frac{E\left( {{h_{t}^{H}w_{t}}}^{2} \right)}{E\left( {{h_{t}^{H}{\overset{\_}{g}}_{t}}}^{2} \right)}},} & (15)\end{matrix}$

denotes the processing gain from the vector quantization unit 28. In theopen-loop predictive vector quantization design approach, the optimalpredictor unit 32 is designed to provide the maximum signal-to-noiseratio level at its output.

With this background, we can now design the vector predictor unit 22modeled by equation (9). In this specification, we use a simple firstorder linear predictor model due for simplicity. The best estimate ofthe current channel direction based on previous channel input is simplygiven by:

{tilde over (g)} _(t) S(g _(t−1))=H _(ouse)(g _(t−1))b,  (16)

where b denotes an N_(t)×1 column vector with unit norm /b/=1. Similarto equation (10), the physical implication of equation (16) is that thebest estimate of the channel direction for the current frame is a linearcombination of the channel direction of the previous frame and itsorthogonal compliment, and the vector b contains the linearcoefficients. For the simple model in equation (16), the optimal linearcoefficients b₁; . . . ; b_(Nt) can be obtained through the maximizationof signal-to-noise ratio predictor. In particular, we have the followingresult:

The linear coefficients that accomplish the highest predictorsignal-to-noise ratio are give by b=[1 0 . . . 0]^(T). Moreover, thepredictor output signal-to-noise ratio level is given by Theorem 1

SNR_(predictor)=1+α²(N _(t)−1).

Based on equation (16), Theorem 1, and the fact that g_(t−1) is thefirst column of H_(ouse)(g_(t−1)), the predictor output is given by{tilde over (g)}_(t)=g_(t−1), i.e., the predictor unit is simplifiedinto a delay unit. It is very difficult to quantify the vectorquantization processing gain in equation (14) analytically, However, itsvalue can be simply lower bounded. The vector quantization processinggain in equation (14) is lower bounded as G_(VQ)≧1.

Combining the above results, we reach a signal-to-noise ratio lowerbound for the open-loop predictive vector quantization system:

$\begin{matrix}{{S\; N\; R} = {{\frac{E_{s}}{\sigma^{2}}S\; N\; R_{predictor}G_{VQ}} \geq {\frac{E_{s}}{\sigma^{2}}\left( {1 + {\alpha^{2}\left( {N_{t} - 1} \right)}} \right)}}} & (18)\end{matrix}$

The signal-to-noise ratio lower bound in equation (18) has severalimportant implications. First, unlike the memory-less vectorquantization beamforming scheme, the signal-to-noise ratio of thepredictive vector quantization scheme is a function of the fadingparameter α. A larger α, or equivalently, a slower fading speed, meanshigher system signal-to-noise ratio. This result also agrees with thecommon intuition that a good beamforming scheme should attain highergain at slower fading environment.

Secondly, at slow fading speed, i.e. the α→1 scenarios, thesignal-to-noise ratio level from the predictive vector quantizationsystem is much larger than the signal-to-noise ratio upper bound fromthe memory-less beamformers. In addition, most of the highsignal-to-noise ratio is obtained from the linear predictor unit 32,which does not exist in the memory-less beamformers. This result closelyresembles the other predictive vector quantization systems in the areasof voice and image coding, where the prediction unit providessignificant amount of processing gains.

The predictive vector quantization beamforming scheme of the illustratedembodiment has taken advantage of the inter-frame correlation. So far,we have designed the optimal predictor unit using an open-loop designapproach. The remaining problem is to design the codebook for the vectorquantization unit 28 in FIG. 2 b using a generalized Lloyd algorithm. Todo this, we adopt a closed-loop vector quantization adaptive codebookalgorithm. To clarify the presentation, the detailed Lloyd trainingalgorithm for the closed-loop vector quantization codebook is summarizedin Table I.

TABLE 1 CLOSED-LOOP PVQ TRAINING ALGORITHM Step 1: Initialization:Prepare random training sequence {h_(t), t = 1, . . . , T}; Generateinitial codebook C₁ using the result from Section III-C; Defineconvergence threshold ψ. Set D₁ = ∞, k = 1. Step 2: Feed the trainingsequence into the PVQ beamformer in FIG. 2. The residue signal ē_(t) isquantized using the current codebook C_(k), and the output from the VQunit is given by {circumflex over (v)}_(t) = arg min_(∀c) _(i) _(⊂C)_(k) d(ē_(t), c_(i)), where the residue signal is given by ē_(t)H_(ouse) ^(H)(ĝ_(t−1))h_(t), and the reconstructed channel direction (orbeamforming weight) is ĝ_(t) = H_(ouse)(ĝ_(t−1)){circumflex over(v)}_(t). $\quad\begin{matrix}{{{{Calculate}\mspace{14mu} D_{k}} = {{\Sigma_{t = 1}^{T}{{\overset{\_}{d}\left( {{\overset{\_}{e}}_{t},{\hat{v}}_{t}} \right)}.\mspace{14mu} {If}}\frac{D_{k - 1} - D_{k}}{D_{k}}} \leq \psi}},{{stop};}} \\{{Otherwise}\mspace{14mu} {goto}\mspace{14mu} {Step}\mspace{14mu} 3.}\end{matrix}$ Step 3: Calculate the centroid of each cluster of trainingsignals that generate the same VQ index, i.e., c_(j) ^(now) = ev(R_(j))= ev(Σ_(∀ē) _(t) _(∈e) _(j) ē_(t) ^(H)ē_(t)), for j = 1, . . . , 2^(N),where ev(•) denotes a operator that returns the eigenvector thatcorresponds to the largest eigenvalue, and θ_(j) ≐ {ē_(t)|{circumflexover (v)}_(t) = c_(j), ∀t ≦ T}. Use c_(j) ^(now) , j = 1, . . . , 2^(N)as the new entries in the new VQ codebook C_(k+1). Set k = k + 1 and goto Step 1.

In the closed-loop approach, the reconstructed channel directionĝ_(t−1), instead of the actual channel direction g_(t−1), will be fedinto the predictor unit 22, thus “closing the loop” as depicted in FIG.2 b. The closed-loop adaptive codebook algorithm is a modified Lloydtraining algorithm. At the initialization stage, an optimal predictorfrom Theorem 1 has already been implemented based on the open-loopanalysis and it will remain unchanged during the training process. Inaddition, an initial vector quantization codebook is also given at thebeginning of the training. At the next step, the residue vectors fromthe residue generator unit 24 will be quantized based on equation (11)using the current codebook in the vector quantization unit 28. Theresidue vectors that belong to a particular codebook entry will begrouped into a cluster, i.e., {tilde over (e)}_(t)ε_(i) if i=min_(c)_(i) _(εC) d({tilde over (e)}_(t), c_(i)) where Θ_(i) denotes thecollection of residue vectors that are quantized using the same codebookentry c_(i). After all the residue vectors have been grouped intoclusters, a new centroid is calculated for each cluster using a minimumdistortion criterion. For example, the new centroid for the ith clusteris calculated as:

$\begin{matrix}{c_{t}^{new} = {{\arg \; {\min\limits_{{\forall{c}} = 1}{\sum\limits_{\forall{{\overset{\sim}{e}}_{t} \in \theta_{i}}}{\overset{\_}{d}\left( {{\overset{\sim}{e}}_{t},c} \right)}}}} = {\arg \; {\min\limits_{{\forall{c}} = 1}{- {\sum\limits_{\forall{{\overset{\sim}{e}}_{t} \in \theta_{i}}}{{{{\overset{\_}{e}}_{t}^{H}c}}^{2}.}}}}}}} & (19)\end{matrix}$

Following the derivation in Xia, the solution to the problem in equation(19) is the eigenvector that corresponds to the largest eigenvalue ofthe matrix R_(i)=Σ∀ē_(t)εΘ_(i) (ē_(t) ^(H)ē_(t)). The group of newcentroid c^(new) _(i); i=1; . . . ; 2^(N) will replace the old codebookentries. As a result, a new vector quantization codebook is generated.This whole training procedure is repeated until a convergence thresholdis met.

Due to the introduction of the feedback loop, the closed-loop designapproach is not guaranteed to be optimal. Since the reconstructedchannel direction instead of the actual channel direction is fed intothe predictor 22 in FIG. 2 b, the predicting gain will be slightly lowerthan the open-loop case. In addition, the above modified Lloyd trainingalgorithm is not guaranteed to reduce the quantization errormonotonically at each step of the iteration. However, the predictivevector quantization predictor will provide a performance very close tothe optimal if the reproduction is of sufficient high quality.Furthermore, the modified Lloyd training algorithm will almost alwaysconverge to a local optimal point in practice. For our predictive vectorquantization beamforming scheme, we have used a relatively largercodebook size 2^(N), thus the near perfect reconstruction assumption isalways satisfied in practice.

Below we will use numerical simulations to demonstrate that the aboveclosed-loop algorithm is effective for normal MISO systems andsubstantial signal-to-noise ratio gain is achieved using the predictivevector quantization beamformer.

Successive Beamforming System

The predictive vector quantization beamformer described above attainsvery high receive signal-to-noise ratios. However, it has a fewdrawbacks. First, the optimal predictive vector quantization codebookneeds to be constructed for each different fading parameter α, which isa prohibitive task for the system designers. Secondly, storing differentcodebooks for different values of a requires a large memory space, whichis also a challenge for practical implementation.

Here, we disclose a novel successive beamforming scheme. This newsuccessive beamforming algorithm significantly reduces theimplementation complexity. In the mean time, it attains nearly the samesignal-to-noise ratio performance as the predictive vector quantizationbeamformer. In certain scenarios, it even surpasses the originalpredictive vector quantization beamformer with better error performance.Note that we have used the notation “successive beamforming” todifferentiate it from the memory-less beamforming or the predictivevector quantization beamforming.

In a successive beamforming system, the codebook is adjusted at eachframe, i.e., the codebook C_(t)={c_(it); i=1; . . . ; 2^(N)} is afunction of time t. Whereas in the memory-less beamformer or thepredictive vector quantization beamformer, the beamforming codebookremains fixed after being constructed. By adjusting the codebook basedon the channel knowledge from the previous frame, the successivebeamformer can benefit from channel correlation and achieve higherprocessing gain. In this section of the specification, our main focus isto develop a codebook adaptation scheme that can keep track of thechannel in an efficient manner and realize a higher receivesignal-to-noise ratio.

Successive Beamforming in Static Fading Cases

Before we design the successive beamforming system scheme for the AR1fading channel, we consider a simple static fading case. In the staticfading scenario, a complex Gaussian MISO channel is realized at thefirst frame, this Gaussian channel has zero-mean and simple covariancecov(h_(i); h_(j))=δ_(ij). Once the MISO channel is realized, it remainsfixed for all the subsequent frames. Obviously, the memory-lessbeamformers in Lemma 1 can be used to generate the transmit weight forthe first frame. At each subsequent frame, new feedback bits reach thetransmitter 16. A successive refinement beamforming scheme can use theseadditional feedback bits to further enhance the signal-to-noise ratioperformance. However, the memory-less beamformers in Lemma 1 cannotsolve this successive refinement problem.

In what follows, we study the successive beamforming system problem inthe static fading scenario. Through the analysis on this simple case, weobtain some useful knowledge on the signal-to-noise ratio performance aswell as the convergence behavior of the successive beamformer. Asillustrated in Lemma 1, after applying the memory-less beamformer to thefirst frame, the system signal-to-noise ratio is upper bounded by

${S\; N\; R} \leq {\frac{E_{s}}{\sigma^{2}}{\left( {N_{t} - {\left( {N_{t} - 1} \right)2^{\frac{- N}{N_{t} - 1}}}} \right).}}$

The upper bound can be reached if and only if

$\begin{matrix}{{\Omega^{N_{t}} = {\bigcup\limits_{i = 1}^{2^{N}}{S_{c_{i}}(\beta)}}},} & (20)\end{matrix}$

where c_(i); i=1; . . . 2^(N) denote the 2^(N) transmit weights in thememory-less beamformer codebook, β=(2)^(−N/(N) ^(t) ⁻¹⁾ and S_(c) _(i)(z)

{g|d_(c) ²(c_(i), g)≦z} denotes a spherical cap centered around thedirection c_(i) with radius z. In other words, the surface of the unithypersphere can be covered by 2^(N) equal sized non-overlappingspherical caps each having radius β. Usually, the signal-to-noise ratioupper bound in Lemma 1 cannot be met except for the trivial N_(t)=2 andN=1 case. However, it has been demonstrated that a good beamformeralways accomplishes a performance very close to this upper bound.Generally, the bound in Lemma 1 is very tight and can be used toapproximate the performance of the actual Grassmannian beamformers.

To simplify our derivation, we assume that this upper bound is alwaysmet at the end of the first frame. Under this assumption, thedistribution of the channel direction conditioned on the firstbeamforming vector w₁ will be a uniform distribution on the sphericalcap S_(w) _(i) (β) i.e., at the second frame, the channel directionsatisfies g˜U(S_(w) _(i) (β)).

At the second frame, a successive beamforming system scheme should havea new codebook C₂={c_(i2); 1≦i≦2^(N)}) Using this new codebook, a highersignal-to-noise ratio performance can be realized. We address thebeamforming problem on the surface area of a spherical cap instead ofthe surface of the whole unit hypersphere. Lemma 3 is stated as follows.In a static fading scenario, assume a good successive beamformer withcodebook C₂={c_(i2); 1≦i≦2^(N)}, then the average signal-to-noise ratiolevel at the second frame is approximated by:

$\begin{matrix}{{{S\; N\; R_{2}} \approx {\frac{E_{s}}{\sigma^{2}}\left( {N_{t} - {\beta_{2}\left( {N_{t} - 1} \right)}} \right)}},} & (21)\end{matrix}$

where β₂

β(1/2^(N))1/(N ^(t) ⁻¹⁾.

The above derivation can be repeated for the consecutive frames andleads to the following result, which is referenced as Corollary 1. In astatic fading scenario, assume that due to the beamforming effect fromthe (t−1)th frame, the conditional distribution of the channel directionsatisfies g˜U(S_(w) _(t−1) (β_(t−1))), then the average signal-to-noiseratio level at the tth frame from a good successive beamformer withcodebook C₂={c_(it); 1≦i≦2^(N)} can be approximated as:

$\begin{matrix}{{{S\; N\; R_{2}} \approx {\frac{E_{s}}{\sigma^{2}}\left( {N_{t} - {\beta_{2}\left( {N_{t} - 1} \right)}} \right)}},} & (22)\end{matrix}$

where β₂

β(1/2^(N))^(1/(N) ^(t) ⁻¹⁾.

The result in Corollary 1 provides a tight upper bound on the achievablesystem signal-to-noise ratio of a successive beamformer. In whatfollows, we will use this result to quantify the processing gain of anideal successive beamformer.

Steady-State Performance of a Successive Beamformer

In this section, we analyze the steady-state performance of an idealsuccessive beamformer using the AR1 fading model in K. Mukkavilli, et.al. “On beamforming with finite rate feedback in multiple-antennasystems,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2562-2579,October 2003. The main goal of this analysis is to find a proper designcriterion for the successive beamforming in the time-varying channel.

In a constantly changing environment, accurate modeling of thebeamformer behavior is very difficult. Therefore, we focus on thesteady-state performance. To simplify the derivation, we make theapproximation that before the current frame, the MISO channel hasexperienced several frames of static fading, and an ideal successivebeamformer from above has been applied for these frames. As a result ofthe successive beamforming system from the previous frame, the currentchannel direction g_(t) is uniformly distributed on a small sphericalcap W_(w) _(t−i) (β_(steady)), i.e. g_(t)˜U(S_(w) _(t−i) (β_(steady))).The conditional channel amplitude follows the distribution in Xia. Inaddition, the channel direction g_(t) is still independent of thechannel amplitude ρ_(t). As we have observed in the numericalsimulations, these approximations are quite close to the actual AR1fading case. This is because for both the AR1 and static fading models,the successive beamformer tracks the channel very closely. In the steadystate, the actual channel direction is usually concentrated in a smallregion around the previous beamforming weight w_(t−1). Therefore, theconditional distribution of the channel direction is well approximatedby a uniform distribution on a small spherical cap.

Based on the above assumptions and following the derivation in Lemma 3and Corollary 1, we can calculate the steady state signal-to-noise ratioat the beginning of the tth frame as:

$\begin{matrix}{{S\; N\; R_{t}} \approx {\frac{E_{s}}{\sigma^{2}}\left( {N_{t} - {\beta_{steady}\left( {N_{t} - 1} \right)}} \right)}} & (23)\end{matrix}$

To determine the parameter β_(steady), we use a two-step approach. Inthe first step, an ideal successive beamformer from above is employed.The successive beamformer generates the transmit weight w_(t) and itenhances the signal-to-noise ratio level in equation (23) to a higherlevel. At the second step, the channel changes to the t+1th frameaccording to equation (2) regardless of the new beamforming weightw_(t). For the two-step approach, the signal-to-noise ratio adaptationcan be modeled as

SNR_(t+1) =G _(chan) G _(bf)SNR_(t),  (24)

where G_(bf) represents the signal-to-noise ratio gain from successivebeamforming system in the first step, and G_(chan) represents thesignal-to-noise ratio degradation due to channel variation in the secondstep. Based on Corollary 1, the beamforming gain in the first step isapproximated as

$\begin{matrix}{{G_{bf} \approx \frac{N_{t} - {{\overset{\sim}{\beta}}_{steady}\left( {N_{t} - 1} \right)}}{N_{t} - {\beta_{steady}\left( {N_{t} - 1} \right)}}},} & (25)\end{matrix}$

where β _(steady)=β_(steady)(1/2^(N))^(1/(N) ^(t) ⁻¹⁾. We then calculatethe parameter G_(chan). At the second step, the channel adapts to thenext frame according to equation (2), and the average signal-to-noiseratio level at the beginning of the t+1th frame can be calculated as

$\begin{matrix}{\quad\begin{matrix}{{S\; N\; R_{t + 1}} = {\frac{E_{s}}{\sigma^{2}}{E\left( {{h_{t + 1}^{H}w_{t}}}^{2} \right)}}} \\{= {\frac{E_{s}}{\sigma^{2}}\left( {{\alpha^{2}{E\left( {{h_{t}^{H}w_{t}}}^{2} \right)}} + {\left( {1 - \alpha^{2}} \right)w_{t}^{H}{E\left( {x_{t + 1}x_{t + 1}^{H}} \right)}w_{t}}} \right)}} \\{{= {{\alpha^{2}G_{bf}S\; N\; R_{t}} + {\frac{E_{s}}{\sigma^{2}}\left( {1 - \alpha^{2}} \right)}}},}\end{matrix}} & (26)\end{matrix}$

where the second equality is due to equation (2) and the fact thatx_(t+1) is zero-mean Gaussian and independent of h_(t); the thirdequality is due to the fact that E(x_(t+1)x^(H) _(t+1))=I_(Nt). Theresults in equations (26) and (24) imply

$\begin{matrix}{G_{chan} = {\alpha^{2} + {\frac{E_{s}\left( {1 - \alpha^{2}} \right)}{\sigma^{2}G_{bf}S\; N\; R_{t}}.}}} & (27)\end{matrix}$

At the steady state, the signal-to-noise ratio level stays constant,i.e. G_(bf)G_(chan)=1. Combining equations (23), (25), and (27), weobtain

$\begin{matrix}{{\beta_{steady} = \frac{1 - \alpha^{2}}{1 - \frac{\alpha^{2}}{\left( {1/2^{N}} \right)^{\frac{1}{N_{t} - 1}}}}},{and}} & (28) \\{{S\; N\; R_{steady}} = {\frac{E_{s}}{\sigma^{2}}{\left( {N_{t} - \frac{\left( {1 - \alpha^{2}} \right)\left( {N_{t} - 1} \right)}{1 - \frac{\alpha^{2}}{\left( {1/2^{N}} \right)^{\frac{1}{N_{t} - 1}}}}} \right).}}} & (29)\end{matrix}$

Codebook Construction for Successive Beamformer

So far, we have established the steady-state performance of a successivebeamformer in the AR1 fading environment. In the next, we will discusshow to construct a series of codebooks that can realize such asignal-to-noise ratio performance. As outlined in above, the primarygoal of the successive adaptive codebook is to achieve the first-stepprocessing gain in equation (25). In Lemma 3 and Corollary 1, we haveshown the processing gain from an ideal adaptive codebook.Straightforwardly, a good design approach is to minimize the performancedifference between a practical codebook and an ideal codebook.

Towards this end, we introduce an important concept as follows. Theconforming radius z₀ of a codebook C_(t) is defined as the largest valueof z such that the relations S_(c) _(it) (z)∩S_(c) _(jt) (z)= and S_(c)_(it) (z)⊂ S_(w) _(t−1) (β_(steady)) are satisfied simultaneously forand ∀i≠j and i; j=1; . . . ; 2^(N).

The above conforming radius is similar to the overlap radius defined inthe prior art. For a practical codebook, its signal-to-noise ratioperformance is closely related to the value of its conforming radius. Inparticular, we have the following result which is called, Lemma 4. Forany codebook C_(t), it holds that F_(z)(z)={tilde over (F)}_(z)(z) whenz≦z₀, and F_(z)(z)≦{tilde over (F)}_(z)(z) when z>z₀, where z

min_(∀i)d_(c) ²(c_(it), g_(t)), and F_(z)(z), {tilde over (F)}_(z)(z)denote the cumulative distribution functions of z for the practical andideal codebooks, respectively.

As shown in the proof of Lemma 3 and Lemma 4, the performance gapbetween an ideal beamformer and a practical beamformer is caused by thedifference between the two cumulative distribution functions F_(z)(z)and {tilde over (F)}_(z)(z). When z≧z₀, F_(z)(z)<{tilde over(F)}_(z)(z). and thus there will be a performance gap between the twocodebooks. Clearly, a good successive codebook can be obtained bymaximizing its conforming radius as much as possible so that thedifference between F_(z)(z) and {tilde over (F)}_(z)(z) can beminimized. As a result, the performance gap between ideal codebook andpractical codebook will also be minimized.

Based on these arguments, we propose the following design criterion.which we call Proposition 1. The design criterion for a good successivebeamforming system codebook is given by

$\begin{matrix}{C_{t} - {\arg \; {\max\limits_{\forall C_{t}}{z_{0}.}}}} & (30)\end{matrix}$

Similar to the Grassmannian design criterion in Lemma 1, the designcriterion in equation (30) is also an optimization problem. Usually,analytical solutions of these optimization problems are difficult toget. In many cases, researchers have developed unique structures thatprovide sub-optimal solutions for the beamformer design problems.Unfortunately, the design criterion in equation (30) is quite differentfrom the maximum chordal distance design criterion in Lemma 1. The samestructural design approaches in the prior art are not applicable for theoptimization problem in equation (30).

In addition, using numerical methods to solve the optimization problemin equation (30) for each fading parameter α is also a difficult task.Similar to the predictive vector quantization beamformer, it alsorequires a large amount of storage space. In practical implementation,it is highly desirable to reduce the number of codebooks for easystorage.

Another critical concern with the successive beamforming system is thatthe codebooks must be updated synchronically at both the transmitter 16and receiver 18 at each frame. If the codebooks on the two sides aremismatched, serious decoding errors will occur.

Overall, in the case of the successive beamforming systems, there areseveral highly desirable features for practical implementation. Thesefeatures include a low complexity construction algorithm, small storagerequirement, and synchronized adaptation at both the transmitter 16 andreceiver 18. In what follows, we introduce a novel systematic codebookgeneration strategy. The proposed scheme will produce a series ofcodebooks as a suboptimal solution for the problem in equation (30).Moreover, this scheme possesses all the above desirable features, thusit is very promising for practical implementation.

We define Proposition 2 as follows. At the tth frame, the successivebeamforming system codebook is generated as:

C _(t) ={c _(it) =H _(ouse)(w _(t−1))[ηe ₁+√{square root over(1−η²)}f_(i)],1≦i≦2^(N)}  (31)

where f_(i)=[0f_(i)]^(T), i=1, . . . , 2^(N) are N_(t)×1 column vectors,{circumflex over (f)}_(i)

[f₁ . . . f_(i() _(t) ⁻¹⁾]^(T) i=1, . . . , 2^(N) are constant(N_(t)−1)×1 column vectors with unit norm. The term η is a scalarparameter and its value is given by

$\begin{matrix}{{\eta = \sqrt{1 - \frac{\beta_{steady}}{\left( {1 + \sqrt{\frac{1 - \xi_{\max}}{2}}} \right)^{2}}}},} & (32)\end{matrix}$

and ξ_(max)

max_(∀i, j) Re({circumflex over (f)}_(i) ^(H){circumflex over (f)}_(j).

In what follows, we show that the codebook generation strategy inequation (31) provides all the aforementioned desirable properties.These features justify the design approach in Proposition 2.

Universal Codebook for all Fading Scenarios:

As demonstrated in equation (31), only a single codebook {circumflexover (F)}={{circumflex over (f)}_(i), i=1, . . . , 2^(N)} is requiredfor the successive beamforming procedure. This codebook consists of2^(N) constant unit norm column vectors. Unlike the predictive vectorquantization beamformer, the present embodiment requires very littlestorage space. In a practical implementation, the codebook {circumflexover (F)} is designed offline and stored at both the transmitter 16 andreceiver 18. For a different fading parameter α, the codebook C_(t) canbe simply derived by adjusting the scalar parameter η. Note that theparameter η is a function of β_(steady), which in turn is uniquelydetermined by the fading parameter α through equation (28).

2) Synchronization without Extra Feedback:

The codebook adaptation procedure in equation (31) is uniquely definedat both the transmitter 16 and the receiver 18. This is because at thetth frame, both the transmitter 16 and receiver 18 know the transmitweight w_(t−1) from the previous frame. The householder matrixH_(ouse)(w_(t−1)) is also uniquely defined. Thus both the transmitter 16and receiver 18 can update H_(ouse)(w_(t−1)) simultaneously. Inaddition, the parameter η is determined through negotiation between thetransmitter 16 and receiver 18, and the same universal codebook{circumflex over (F)} is known on both sides. Based on the structure inequation (31), both sides 16 and 18 of the wireless link can updatetheir beamforming codebooks C_(t) simultaneously without any extrafeedback.

3) Simple Codebook Construction on Universal Codebook {circumflex over(F)}:

To derive the optimal universal codebook {circumflex over (F)}, we firstnote an important result regarding the conforming radius of thesuccessive codebook C_(t). It is designated as Theorem 2. In a slowfading environment, if the codebook is designed according to thestrategy in equation (31), then the conforming radius of C_(t) is givenby

$\begin{matrix}{z_{0} = {\frac{\beta_{steady}\left( {1 - \xi_{\max}} \right)}{2\left( {1 + \sqrt{\frac{1 - \xi_{\max}}{2}}} \right)^{2}}.}} & (33)\end{matrix}$

Based on definition, the feasible range of ξ_(max) is ξ_(max)ε[−1, 1].In addition, it can be easily verified that z₀ is a monotonicallydecreasing function of ξ_(max). Combining these results and Proposition1, we obtain the design criterion for the universal codebook {circumflexover (F)}:

$\begin{matrix}{\hat{F} = {{\arg \; {\min\limits_{\forall\hat{F}}\xi_{\max}}} = {\arg \; {\min\limits_{\forall\hat{F}}{\max\limits_{{1 \leq i},{j \leq N}}{{{Re}\left( {{\hat{f}}_{i}^{H}{\hat{f}}_{j}} \right)}.}}}}}} & (34)\end{matrix}$

The simple design criterion in equation (34) suggests that the codebook{circumflex over (F)} is independent of the fading parameter α. Thisfeature is very convenient for a practical implementation. The codebook{circumflex over (F)}can be constructed offline regardless of the actualfading. Once in the operational mode, the actual codebook C_(t) can beeasily derived based on equation (31). The optimization problem inequation (34) can be solved using standard numerical methods. In thisembodiment, we generate the universal codebook {circumflex over (F)}using the fminmax( ) function in the optimization toolbox of MATLAB.

In the above disclosure, we have established a method of designing asuccessive beamformer, along with its primary implementation advantages.The proposed successive beamforming system is based on a slow fadingassumption. However, as demonstrated below, the codebook constructedbased on equation (34) yields superior performance at both high and lowfading speeds.

Numerical Simulations and Implementation Issues

To illustrate the performance of the illustrated beamformingembodiments, we present Monte Carlo simulations for several differentMISO systems. In these simulations, we assume that the wireless systemis operating at a carrier frequency equal to 2 GHz. In addition, afeedback channel with 4500 bps bandwidth has been adopted. For acodebook with 2^(N) entries, the feedback interval is N=4500 second. Inthe bit error rate (BER) simulations, we use the biphase shift keying(BPSK) constellation which transmits at rate 1 bit/s/Hz. In thesesimulations, we assume that the feedback channel is error free, andthere is one frame feedback delay between the transmitter 16 andreceiver 18. For fair comparison, all the beamforming schemes have equaltransmit power. It is to be understood that each of these parameterscould be varied according to the illustration used and none should beregarded as a limitation of the scope of the invention.

Comparing on Average Receive Signal-to-Noise Ratio:

We have implemented both the predictive vector quantization and thesuccessive beamforming embodiments disclosed above. For each embodiment,we assume E_(s)/σ²=1 and then use Monte Carlo simulations to examine theactual receive signal-to-noise ratio for various fading parameters α. Wealso calculate the theoretical steady-state signal-to-noise ratio valueusing equation (29). The results are summarized in the graph of FIG. 3where the signal-to-noise ration is graphed against mobile speed for asystem with N_(t)=3; N=8, and in the graph of FIG. 4 for a system withN_(t)=4; N=16. The theoretical signal-to-noise ratio upper bound for thememory-less Grassmannian beamformer (MGB) in Lemma 1 is also included inboth graphs of FIGS. 3 and 4.

As can be seen from the graphed results, the predictive vectorquantization (PVQ) embodiment accomplishes the best signal-to-noiseratio level among the different beamforming embodiments. This resultagrees with the common observation that codebooks designed using Lloydalgorithm normally generate very good signal-to-noise ratio performance.The signal-to-noise ratio performance of the successive beamformingsystem (SBF) embodiment comes very close to that of the predictivevector quantization embodiment. In most cases, the difference is within0.1 dB. This result confirms that the successive beamforming embodiment,though not necessarily optimal, can deliver an average signal-to-noiseratio nearly as good as the numerical vector quantization beamformingembodiment, and this high signal-to-noise ratio performance is obtainedusing a much simpler implementation.

In addition, the theoretical steady-state signal-to-noise ratio inequation (29) and the actual signal-to-noise ratios from the successivebeamforming embodiments are very close. Their difference is especiallysmall at low mobile speeds within 0.1 dB for speed ≦30 km/h. For fastfading cases, the discrepancy increases slightly. This increasingdifference could be due to the fact that our successive beamformingembodiment is designed based on a low fading speed assumption.

In general, the analytical signal-to-noise ratio in equation (29)closely predicts the actual signal-to-noise ratio performance of thesuccessive beamforming embodiment. Finally, both the predictive vectorquantization and successive beamforming embodiments outperform thememoryless Grassmannian beamformer (MGB) embodiment. At slow fadingscenario, more than 1.3 dB signal-to-noise ratio gain is observed.

Bit Error Rate (BER) Performance:

We now examine the BER performance for four different transmitbeamforming embodiments: the MGB embodiment, the predictive vectorquantization embodiment, the successive beamforming system, and an idealbeamformer which uses perfect channel knowledge at the transmitter 16.The BER-signal-to-noise ratio curves for N_(t)=3; N=3 case and N_(t)=4;N=4 cases are depicted in the graphs of FIGS. 5 and 6, respectively. Ascan be seen from these figures, the ideal beamformer with perfectchannel state information provides an BER lower bound for all the otherembodiments. For the beamformers with finite rate feedback, thepredictive vector quantization beamformer provides the best performanceat low signal-to-noise ratios. The successive beamforming embodimentattains nearly the same performance as that of the predictive vectorquantization beamformer. Both are better than the MGB embodiment.However, most of the performance improvement is obtained at the lowsignal-to-noise ratio range. At the high signal-to-noise ratio range,the performance gain from the predictive vector quantization andsuccessive beamforming system embodiments diminishes. In certainscenarios, it is even inferior to the MGB embodiment.

This result is actually not surprising. In this specification, we havefocused on a beamforming embodiment that can maximize the averagereceive power. However, it has been pointed out in the art that themaximum receive power design criterion is only valid for the lowsignal-to-noise ratio environment. At high signal-to-noise ratios, mostof the symbol errors are caused by the few scenarios where the channelis in deep fading. For the AR1 model in equation (2), if the channel isin deep fading, a small adjustment term αx_(t) will result in a largevariation in the channel direction g_(t). For a successive beamformer,its performance at high signal-to-noise ratio will be determined by itsability to track the channel effectively in deep fading. A coarse butfast tracking of the channel is much more effective than a fine but slowtracking channel. The maximization of the average receivesignal-to-noise ratio, on the other hand, will usually result in a finebut slow tracking performance.

To enhance the tracking performance of the successive beamforming systemembodiment at high signal-to-noise ratios, we can simply adjust theparameter η in equation (31). As implied above, the tracking performanceof the successive beamforming system algorithm is determined by theparameter η, which is in turn determined by the value of β_(steady).When the channel is in deep fading, the conditional channel directionwill be distributed on a larger spherical cap S_(w) _(t−1) (β_(steady))For the embodiment in equation (31), the codebook can cover a largerspherical cap by simply using a smaller η. For a fast and coarsetracking, a straightforward approach is to use a smaller η at highsignal-to-noise ratio range. At the extreme case, if η=0, the designcriterion in Proposition 1 will be reduced to the Grassmannianbeamformer design criterion in Lemma 1.

To determine the optimal value of η, we use a numerical approach. TheBER simulations were performed for 1−η in steps of one-quarter of adecade, i.e., in a factor of 1.778 apart. At different signal-to-noiseratio levels, the optimal value for η is obtained through extensive BERsimulations. Such a numerical approach has also been investigated forthe GFB algorithms in the prior art and significant performanceimprovement has been obtained. In a practical implementation, theoptimal values of η are stored in a lookup table. Based on the currentsignal-to-noise ratio and fading speed, the optimal η is calculated atthe receiver 18 and sent back to the transmitter 16. As demonstrated inthe subsequent simulations, a coarse quantization on η will result insignificant performance improvement. In addition, since the fadingparameter α is a relatively slowly changing parameter, the feedback ofthis parameter incurs very little feedback overhead.

Enhanced BER Performance:

The BER as a function of η is investigated through numerical search fordifferent mobile speeds. The results are summarized in the graph ofFIGS. 7 and 8 for N_(t)=3 and N_(t)=4 cases, respectively. Both figuresconfirm that the analytical value of η in equation (32) is only accurateat low signal-to-noise ratios, and a smaller η will result in better BERperformance at high signal-to-noise ratios.

Based on the results in the graphs of FIGS. 7 and 8, we define a verysimple adaptive feedback embodiment. Only two discrete values areallowed to be used for the parameter η for the codebook generationembodiment in equation (31), namely η_(high)=0:9769 for lowsignal-to-noise ratio scenarios and η_(low)=0:8702 for highsignal-to-noise ratio scenarios. These two values correspond to the twovertical lines on the numerical sweep results in FIG. 7 and FIG. 8. Notethat the selection of η_(high) or η_(low) is not unique. These twoheuristic values are chosen here because they yield fairly good BERperformance. In addition, the receiver 18 also maintains a lookup tablewhich stores a series of signal-to-noise ratio threshold valuesτ(v_(mb)). The values of τ(v_(mb)) are summarized in Table II fordifferent mobile speed and different system settings.

TABLE II SNR THRESHOLDS FOR THE QUANTIZATION OF η Mobile Speed SNRThreshold (dB) N_(t) = 3, N = 3 case 10 km/h 14 30 km/h 8 50 km/h −2N_(t) = 4, N = 4 case 10 km/h 12 30 km/h 3 50 km/h −6 When SN R ≧Threshold, η = η_(low) = 0.8702 When SN R < Threshold, η = η_(high) =0.9769

At mobile speed v_(mb), if SNR<τ(v_(mb)), η_(high) will be utilized togenerate the beamforming codebook in equation (31). Otherwise, η_(low)will be selected. In a practical implementation, the receiver 18monitors the signal-to-noise ratio and fading speed in real time. Oncethe fading environment changes significantly, one bit of feedback on theselection of η will be sent back to the transmitter 16. As outlined fromthe above discussion, the proposed adaptive embodiment incurs verylittle system overhead.

The BER curves for the adaptive successive beamforming embodiment isdepicted in FIGS. 9 and 10 for the N_(t)=3; N=3 and N_(t)=4; N=4 cases,respectively. Both embodiments show significant improvement on the BERperformance compared to the original successive beamforming embodimentin FIGS. 5 and 6. Actually, by using the η value in Table II, thetracking performance of the adaptive successive beamforming algorithm isnow even better than the original predictive vector quantizationbeamformer. As a result, the adaptive successive beamforming algorithmoutperforms the MGB embodiment as well as the predictive vectorquantization embodiment at high signal-to-noise ratios.

Comparison with Stochastic Gradient Adaptation Algorithm:

Besides the disclosed successive beamforming embodiment, anotherimportant adaptive beamforming embodiment is the gradient feedback (GFB)algorithm used in the prior art. The GFB embodiment uses a typicaladaptive filtering approach. Based on the same AR1 fading model, the GFBalgorithms have shown superior performance compared to other vectorquantization based beamforming algorithms. To compare the bit errorperformance of our successive beamformer with that of the GFB algorithm,we carried out a series of experiments for a system with parametersN_(t)=4 and N=4. In the latest orthogonal projection based GFB algorithm(GFBOP) of the prior art, two feedback bits are used to select one outof several random perturbations on the transmit weight. Meanwhile, oursuccessive beamforming embodiment requires N=4 feedback bits per update.Since the number of feedback bits are different, the adaptation intervalof the different beamforming embodiment is adjusted for fair comparison,i.e., the GFBOP embodiment and our successive beamforming systemembodiments are updated every two and four frames, respectively. The keyadaptation parameters α=[0:9970; 0:9704] and β=0:316 are adopted for theGFBOP embodiment.

For our successive beamforming system embodiment, the key codebookgeneration parameter is set to be η=0:8702 for all the differentsignal-to-noise ratios. The BER-signal-to-noise ratio curves for theabove beamforming embodiments are summarized in the graph of FIG. 11. Atthe slow fading case (α=0:9970 case), the GFB algorithm comes close tothe successive beamforming embodiment. At the faster fading case(α=0:9704 case), the GFB embodiment becomes inferior to our successivebeamforming embodiment.

The performance gain from our successive beamforming embodiment is notsurprising, since the weight update in the GFB algorithm is based on therandom perturbation on the transmit weight. Therefore, the beamformingweight is not guaranteed to follow the steepest descent path at eachadaptation. On the other hand, our successive beamforming algorithmalways chooses the most favorite direction from the 2^(N) transmitweights at each adaptation. In general, our algorithm follows thechannel in a more efficient manner and hence it works better for fasterfading cases. It is also worth mentioning that, the GFB algorithmrequires that the randomly generated weight perturbation beingtransmitted on the forward pilot link. Therefore, the GFB embodimentconsumes considerable amount of bandwidth and power resources at thetransmitter 16. Our successive beamforming embodiment, on the otherhand, does not have such requirement. Therefore, the successivebeamforming system algorithm enjoys considerable implementationadvantages over the GFB embodiment.

In summary, we have disclosed above a beamformer design problem based ona simple AR1 fading model. To exploit the time domain correlation in thechannel fading, we associate the beamforming problem with a predictivevector quantization problem. We have constructed a predictive vectorquantization beamformer which accomplishes superior power delivery.However, the predictive vector quantization beamformer has a highimplementation complexity. To simplify the implementation, we havedeveloped a novel successive beamforming algorithm as well. The newalgorithm is based on a universal adaptive codebook and works at anyfading environment. The signal-to-noise ratio performance of thealgorithm has been studied analytically and has been found to match thenumerical simulation. Monte Carlo simulations have confirmed that theproposed successive beamforming embodiment attains nearly the samesignal-to-noise ratio performance as the predictive vector quantizationbeamformer, yet it enjoys much simpler implementation. The trackingperformance of the proposed successive beamforming system algorithm isfurther enhanced for high signal-to-noise ratio scenarios. Finally,comparison with the memory-less Grassmannian beamformer and thestochastic gradient beamformers confirm that our new beamformer providesthe best performance over various fading scenarios.

Many alterations and modifications may be made by those having ordinaryskill in the art without departing from the spirit and scope of theinvention. Therefore, it must be understood that the illustratedembodiment has been set forth only for the purposes of example and thatit should not be taken as limiting the invention as defined by thefollowing invention and its various embodiments.

Therefore, it must be understood that the illustrated embodiment hasbeen set forth only for the purposes of example and that it should notbe taken as limiting the invention as defined by the following claims.For example, notwithstanding the fact that the elements of a claim areset forth below in a certain combination, it must be expresslyunderstood that the invention includes other combinations of fewer, moreor different elements, which are disclosed in above even when notinitially claimed in such combinations. A teaching that two elements arecombined in a claimed combination is further to be understood as alsoallowing for a claimed combination in which the two elements are notcombined with each other, but may be used alone or combined in othercombinations. The excision of any disclosed element of the invention isexplicitly contemplated as within the scope of the invention.

The words used in this specification to describe the invention and itsvarious embodiments are to be understood not only in the sense of theircommonly defined meanings, but to include by special definition in thisspecification structure, material or acts beyond the scope of thecommonly defined meanings. Thus if an element can be understood in thecontext of this specification as including more than one meaning, thenits use in a claim must be understood as being generic to all possiblemeanings supported by the specification and by the word itself.

The definitions of the words or elements of the following claims are,therefore, defined in this specification to include not only thecombination of elements which are literally set forth, but allequivalent structure, material or acts for performing substantially thesame function in substantially the same way to obtain substantially thesame result. In this sense it is therefore contemplated that anequivalent substitution of two or more elements may be made for any oneof the elements in the claims below or that a single element may besubstituted for two or more elements in a claim. Although elements maybe described above as acting in certain combinations and even initiallyclaimed as such, it is to be expressly understood that one or moreelements from a claimed combination can in some cases be excised fromthe combination and that the claimed combination may be directed to asubcombination or variation of a subcombination.

Insubstantial changes from the claimed subject matter as viewed by aperson with ordinary skill in the art, now known or later devised, areexpressly contemplated as being equivalently within the scope of theclaims. Therefore, obvious substitutions now or later known to one withordinary skill in the art are defined to be within the scope of thedefined elements.

The claims are thus to be understood to include what is specificallyillustrated and described above, what is conceptionally equivalent, whatcan be obviously substituted and also what essentially incorporates theessential idea of the invention.

1. An improvement in a method of transmit beamforming between atransmitter and a receiver for a time varying fading channel comprisingperforming transmit beamforming using less than complete knowledge ofthe previous fading blocks by beamforming an adaptive codebook of acurrent fading block with each time frame.
 2. The improvement of claim 1where performing transmit beamforming comprises performing a successivebeamforming algorithm.
 3. The improvement of claim 1 where performingtransmit beamforming comprises performing a vector quantizationbeamforming algorithm.
 4. The improvement of claim 1 further comprisingdetermining a fading parameter α at least the transmitter or receiver bymonitoring a mobile Doppler frequency.
 5. The improvement of claim 1where performing transmit beamforming using less than complete knowledgeof the previous fading blocks by beamforming a adaptive codebook of acurrent fading block with each time frame comprises generating a bestestimated channel direction {tilde over (g)}_(t) based on the pastchannel inputs.
 6. The improvement of claim 1 where generating a bestestimated channel direction {tilde over (g)}_(t) based on the pastchannel inputs comprises generating an output {tilde over(g)}_(t)=S(g_(t−1)) where S(▪) denotes a predicting function, {tildeover (g)}_(t) denotes a best estimate of the current channel directionbased on the previous channel direction g_(t−1).
 7. The improvement ofclaim 1 where performing transmit beamforming comprises generating anoriginal channel vector h_(t), converting the original channel vectorh_(t) into a residue vector {tilde over (e)}_(t) in a residue generatorin a closed-loop encoder, where {tilde over (e)}_(t)=H_(ouse)({tildeover (g)}_(t))h_(t), where H_(ouse) is a Householder matrix.
 8. Theimprovement of claim 7 where performing transmit beamforming comprisesquantizing the residue vector {tilde over (e)}_(t) by a vectorquantization block in a closed-loop encoder to generate a quantizedresidue vector is denoted as {circumflex over (v)}_(t) based on aminimum distortion criterion using a distance metric d(, ), where{circumflex over (v)}_(t)=arg min_(c) _(t) _(εC) d({tilde over (e)}_(t),c_(i)), and C is a codebook, C={c₁, . . . , c_(N)}.
 9. The improvementof claim 1 where performing transmit beamforming comprises generating areconstructed channel direction g, in a reconstruction unit aclosed-loop encoder according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t).
 10. The improvement of claim 6 whereperforming transmit beamforming comprises generating an original channelvector h_(t), converting the original channel vector h_(t) into aresidue vector {tilde over (e)}_(t) in a residue generator in aclosed-loop encoder, where {tilde over (e)}_(t)=H_(ouse)({tilde over(g)}_(t))h_(t) where H_(ouse) is a Householder matrix, generating areconstructed channel direction ĝ_(t) in a reconstruction unit aclosed-loop encoder according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t), and feeding back {tilde over (g)}_(t) thebest estimate of the current channel direction based on the previouschannel direction g, to the residue generator and to the reconstructionunit in the closed loop encoder.
 11. The improvement of claim 1 whereperforming transmit beamforming comprises receiving the feedback signal,i_(t), inverse quantizing the feedback signal, i_(t), to generate aquantized residue vector is denoted as {circumflex over (v)}_(t) basedon a minimum distortion criterion using a distance metric d(, ), where{circumflex over (v)}_(t)=arg min_(c) _(i) _(εC) d({tilde over (e)}_(t),c_(i)), and C is a codebook, C={c₁, . . . c_(N)}, generating areconstructed channel direction ĝ_(t) in a reconstruction unit aclosed-loop decoder according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t), and feeding back {tilde over (g)}_(t) thebest estimate of the current channel direction based on the previouschannel direction g_(t−1) to the reconstruction unit in the closed loopdecoder.
 12. The improvement of claim 8 where quantizing the residuevector {tilde over (e)}_(t) by a vector quantization block in aclosed-loop encoder comprises generating a closed-loop vectorquantization codebook using a Lloyd training algorithm.
 13. Theimprovement of claim 12 where generating a closed-loop vectorquantization codebook using a Lloyd training algorithm comprises aroutine, the routine comprising initializing an optimal predictor basedon the open-loop analysis, generating an initial vector quantizationcodebook, quantizing the residue vectors from the residue generator unitusing a current codebook in the vector quantization unit, grouping theresidue vectors that belong to a particular codebook entry will begrouped into a cluster, i.e., {tilde over (e)}_(t)εΘ_(i) if i=min_(c)_(i) _(εC) d({tilde over (e)}_(t), c_(i)) where Θ_(i) denotes thecollection of residue vectors that are quantized using the same codebookentry c_(i), determining a new centroid using a minimum distortioncriterion, replacing prior codebook entries with, the new centroidc^(new) _(i); i=1; . . . ; 2^(N) to provide a new vector quantizationcodebook; and repeating the routine until a convergence threshold issatisfied.
 14. The improvement of claim 13 where determining a newcentroid using a minimum distortion criterion comprises generating newcodebook entries, c_(i) ^(new) as modeled by$c_{i}^{new} = {{\arg \; {\min\limits_{{\forall{c}} = 1}{\sum\limits_{\forall\; {{\overset{\sim}{e}}_{t} \in \Theta_{i}}}{\overset{\_}{d}\left( {{\overset{\sim}{e}}_{t},c} \right)}}}} = {\arg \; {\min\limits_{{\forall{c}} = 1}{- {\sum\limits_{\forall{{\overset{\sim}{e}}_{t} \in \Theta_{i}}}{{{{\overset{\sim}{e}}_{t}^{\overset{\_}{H}}c}}^{2}.}}}}}}$by solving for an eigenvector that corresponds to the largest eigenvalueof the matrixR _(i)=Σ∀ē_(t)εΘ_(i)({tilde over (e)} _(t) ^(H) ē _(t)).
 15. Theimprovement of claim 1 where blocks are transmitted and received inframes and where performing transmit beamforming using less thancomplete knowledge of the previous fading blocks by beamforming aadaptive codebook of a current fading block with each time framecomprises adjusting the codebook at each frame.
 16. The improvement ofclaim 15 where channel fading is static and where adjusting the codebookat each frame comprises generating a codebook with an averagesignal-to-noise ratio level at the t^(th) frame with codebookC₂={c_(it); 1≦i≦2^(N)} approximated as:${{S\; N\; R_{2}} \approx {\frac{E_{s}}{\sigma^{2}}\left( {N_{t} - {\beta_{2}\left( {N_{t} - 1} \right)}} \right)}},$where β₂

β(1/2 ^(N))^(1/(N) ^(t) ⁻¹⁾.
 17. The improvement of claim 15 wherechannel fading is time varying and where adjusting the codebook at eachframe at steady state conditions comprises generating a codebook with anaverage signal-to-noise ratio level at the t^(th) frame with codebookC_(i)={c_(it); 1≦i≦2^(N)} approximated as:${S\; N\; R_{steady}} = {\frac{E_{s}}{\sigma^{2}}{\left( {N_{t} - \frac{\left( {1 - \alpha^{2}} \right)\left( {N_{t} - 1} \right)}{1 - \frac{\alpha^{2}}{\left( {1/2^{N}} \right)^{\frac{1}{N_{t} - 1}}}}} \right).}}$18. The improvement of claim 15 where channel fading is time varying andwhere adjusting the codebook at each frame at steady state conditionscomprises generating a codebook with an average signal-to-noise ratiolevel at the t^(th) frame with codebook C_(i)={c_(it); 1≦i≦2^(N)}approximated as:C _(t) ={c _(it) =H _(ouse)(w _(t−1))[ηe ₁+√{square root over(1−η²)}f_(i)],1≦i≦2^(N)} where f_(i)=[0{circumflex over (f)}_(i)]^(T),i=1, . . . , 2^(N) are N_(t)×1 column vectors, {circumflex over (f)}

[f_(i1) . . . f_(i(N) _(t) ⁻¹⁾]^(T) i=1, . . . , 2^(N) are constant(N_(t)−1)×1 column vectors with unit norm, where η is a scalar parameterand its value is given by${\eta = \sqrt{1 - \frac{\beta_{steady}}{\left( {1 + \sqrt{\frac{1 - \xi_{\max}}{2}}} \right)^{2}}}},$and ξ_(max)

max_(∀i,j) Re({tilde over (f)}_(i) ^(H){circumflex over (f)}_(j)). 19.The improvement of claim 15 where channel fading is time varying andwhere adjusting the codebook at each frame at steady state conditionscomprises simultaneously updating H_(ouse)(w_(t−1)) at both thetransmitter and receiver.
 20. The improvement of claim 15 where channelfading is time varying and where adjusting the codebook at each frame atsteady state conditions comprises simultaneously generating an identicaluniversal codebook {circumflex over (F)} in the transmitter andreceiver.
 21. The improvement of claim 18 where generating a codebookcomprises generating a universal codebook satisfying the criterion:$\hat{F} = {{\arg \; {\min\limits_{\forall\hat{F}}\xi_{\max}}} = {\arg \; {\min\limits_{\forall\hat{F}}{\max\limits_{{1 \leq i},{j \leq N}}{{{Re}\left( {{\hat{f}}_{i}^{H}{\hat{f}}_{j}} \right)}.}}}}}$22. A method of closed-loop encoding for transmitting a beamformingsignal using less than complete knowledge of the previous fading blocksby beamforming a adaptive codebook of a current fading block with eachtime frame comprising: generating an original channel vector h_(t),converting the original channel vector h_(t) into a residue vector{tilde over (e)}_(t) in a residue generator in a closed-loop encoder,where {tilde over (e)}_(t)=H_(ouse)({tilde over (g)}_(t))h_(i), whereH_(ouse) is a Householder matrix; quantizing the residue vector {tildeover (e)}_(t) by a vector quantization block in a closed-loop encoder togenerate a quantized residue vector is denoted as {circumflex over(v)}_(t) based on a minimum distortion criterion using a distance metricd(, ), where {circumflex over (v)}_(t)=arg min_(c) _(i) _(εC) d({tildeover (e)}_(t), c_(i)), and C is a codebook, C={c₁, . . . , c_(N)} and togenerate a feedback signal it from the vector quantization block;generating a reconstructed channel direction ĝ_(t) in a reconstructionunit a closed-loop encoder according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t); generating a best estimated channeldirection {tilde over (g)}_(t) based on the past channel inputs bygenerating an output {tilde over (g)}_(t)=S(g_(t−1)) where S() denotesa predicting function, {tilde over (g)}_(t) denotes a best estimate ofthe current channel direction based on the previous channel directiong_(t−1); and feeding back {tilde over (g)}_(t) the best estimate of thecurrent channel direction based on the previous channel directiong_(t−1) to the residue generator and to the reconstruction unit in theclosed loop encoder.
 23. A method for closed-looped decoding byreceiving a beamforming signal using less than complete knowledge of theprevious fading blocks by beamforming a adaptive codebook of a currentfading block with each time frame comprising: receiving the feedbacksignal, i_(t); inverse quantizing the feedback signal, i_(t), togenerate a quantized residue vector is denoted as {circumflex over(v)}_(t) based on a minimum distortion criterion using a distance metricd(, ), where {circumflex over (v)}_(t)=arg min_(c) _(t) _(εC) d({tildeover (e)}_(t); c_(i)), and C is a codebook, C={c₁, . . . , c_(N)};generating a reconstructed channel direction ĝ_(t) in a reconstructionunit a closed-loop decoder according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t); and feeding back {tilde over (g)}_(t) thebest estimate of the current channel direction based on the previouschannel direction g_(t−1) to the reconstruction unit in the closed loopdecoder.
 24. The improvement of claim 1 where performing transmitbeamforming comprises constructing a time-varying beamforming codebookbased on less than complete knowledge from a previous fading block. 25.The improvement of claim 1 where performing transmit beamformingcomprises selecting a transmit weight from a previous fading block tocarry out transmit beamforming on a current fading block with each timeframe.
 26. The improvement of claim 1 where performing transmitbeamforming comprises performing a successive beamforming embodimentwith only a single codebook required in both the transmitter andreceiver without multiple codebooks for different fading speeds.
 27. Animprovement in a wireless system including a transmitter and a receiverwith a time varying fading channel comprising means for performingtransmit beamforming using less than complete knowledge of the previousfading blocks by beamforming a adaptive codebook of a current fadingblock with each time frame.
 28. The improvement of claim 27 where themeans for performing transmit beamforming comprises means for performinga successive beamforming algorithm.
 29. The improvement of claim 27where the means for performing transmit beamforming comprises means forperforming a vector quantization beamforming algorithm.
 30. Theimprovement of claim 27 further comprising means for determining afading parameter α at least the transmitter or receiver by monitoring amobile Doppler frequency.
 31. The improvement of claim 27 where meansfor performing transmit beamforming using less than complete knowledgeof the previous fading blocks by beamforming a adaptive codebook of acurrent fading block with each time frame comprises means for generatinga best estimated channel direction {tilde over (g)}_(t) based on thepast channel inputs.
 32. The improvement of claim 27 where the means forgenerating a best estimated channel direction {tilde over (g)}_(t) basedon the past channel inputs comprises means for generating an output{tilde over (g)}_(t)=S(g_(t−1)) where S() denotes a predictingfunction, {tilde over (g)}_(t) denotes a best estimate of the currentchannel direction based on the previous channel direction g_(t−1). 33.The improvement of claim 27 where the means for performing transmitbeamforming comprises means for generating an original channel vectorh_(t), and a residue generator means for converting the original channelvector h_(t) into a residue vector {tilde over (e)}_(t), where {tildeover (e)}_(t)=H_(ouse)({tilde over (g)}_(t))h_(t), where H_(ouse) is aHouseholder matrix.
 34. The improvement of claim 33 where the means forperforming transmit beamforming comprises vector quantization means forquantizing the residue vector e; to generate a quantized residue vectoris denoted as {circumflex over (v)}_(t) based on a minimum distortioncriterion using a distance metric d(, ), where {circumflex over(v)}_(t)=arg min_(c) _(i) _(εC) d({tilde over (e)}_(t), c_(i)) and C isa codebook, C={c₁, . . . , c_(N)}.
 35. The improvement of claim 27 wherethe means for performing transmit beamforming comprises reconstructionmeans for generating a reconstructed channel direction ĝ_(t) accordingto ĝ_(t)=H_(ouse)({tilde over (g)}_(t)){tilde over (v)}_(t).
 36. Theimprovement of claim 32 where the means for performing transmitbeamforming comprises residue generator means for generating an originalchannel vector h_(t), means for converting the original channel vectorh_(t) into a residue vector {tilde over (e)}_(t), where {tilde over(e)}_(t)=H_(ouse)({tilde over (g)}_(i))h_(t), where H_(ouse) is aHouseholder matrix, reconstruction means for generating a reconstructedchannel direction ĝ_(t) according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t), and means for feeding back {tilde over(g)}_(t) the best estimate of the current channel direction based on theprevious channel direction g_(t−1) to the residue generator means and tothe reconstruction means.
 37. The improvement of claim 27 where themeans for performing transmit beamforming comprises means for receivingthe feedback signal, i_(t), means for inverse quantizing the feedbacksignal, i_(t), to generate a quantized residue vector is denoted as{circumflex over (v)}_(t) based on a minimum distortion criterion usinga distance metric d(, ), where {circumflex over (v)}_(t)=arg min_(c)_(i) _(εC) d({tilde over (e)}_(i), c_(i)), and C is a codebook, C={c₁, .. . , c_(N)}, reconstruction means for generating a reconstructedchannel direction ĝ_(t) according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t), and means for feeding back {tilde over(g)}_(t) the best estimate of the current channel direction based on theprevious channel direction g_(t−1) to the reconstruction means.
 38. Theimprovement of claim 34 where the vector quantization means forquantizing the residue vector {tilde over (e)}_(t) comprises means forgenerating a closed-loop vector quantization codebook using a Lloydtraining algorithm.
 39. The improvement of claim 38 where the means forgenerating a closed-loop vector quantization codebook using a Lloydtraining algorithm comprises means for initializing an optimal predictorbased on the open-loop analysis, means for generating an initial vectorquantization codebook, means for quantizing the residue vectors from theresidue generator means using a current codebook in the vectorquantization means, means for grouping the residue vectors that belongto a particular codebook entry will be grouped into a cluster, i.e.,{tilde over (e)}_(t)εΘ_(i) if i=min_(c) _(i) _(εC) d({tilde over(e)}_(t), c_(i)) where Θ_(i) denotes the collection of residue vectorsthat are quantized using the same codebook entry c_(i), means fordetermining a new centroid using a minimum distortion criterion,replacing prior codebook entries with the new centroid c^(new) _(i);i=1; . . . ; 2^(N) to provide a new vector quantization codebook; andmeans for repeating operational cycles of each of the foregoing meansuntil a convergence threshold is satisfied.
 40. The improvement of claim39 where the means for determining a new centroid using a minimumdistortion criterion comprises generating new codebook entries, c_(i)^(new) as modeled by$c_{i}^{new} = {{\arg \; {\min\limits_{{\forall{c}} = 1}{\sum\limits_{\forall\; {{\overset{\sim}{e}}_{t} \in \Theta_{i}}}{\overset{\_}{d}\left( {{\overset{\sim}{e}}_{t},c} \right)}}}} = {\arg \; {\min\limits_{{\forall{c}} = 1}{- {\sum\limits_{\forall{{\overset{\sim}{e}}_{t} \in \Theta_{i}}}{{{{\overset{\sim}{e}}_{t}^{\overset{\_}{H}}c}}^{2}.}}}}}}$by solving for an eigenvector that corresponds to the largest eigenvalueof the matrix R_(i)=Σ∀ē_(t)εΘ_(i)(ē_(t) ^(H)ē_(t)).
 41. The improvementof claim 27 where blocks are transmitted and received in frames andwhere the means for performing transmit beamforming using less thancomplete knowledge of the previous fading blocks by beamforming aadaptive codebook of a current fading block with each time framecomprises means for adjusting the codebook at each frame.
 42. Theimprovement of claim 41 where channel fading is static and where themeans for adjusting the codebook at each frame comprises means forgenerating a codebook with an average signal-to-noise ratio level at thet^(th) frame with codebook C₂={c_(it); 1≦i≦2^(N)} approximated as:${{S\; N\; R_{2}} \approx {\frac{E_{s}}{\sigma^{2}}\left( {N_{t} - {\beta_{2}\left( {N_{t} - 1} \right)}} \right)}},$where β₂

β(1/2^(N))^(1/N) ^(t) ⁻¹⁾.
 43. The improvement of claim 41 where channelfading is time varying and where the means for adjusting the codebook ateach frame at steady state conditions comprises means for generating acodebook with an average signal-to-nolse ratio IeVCi at the t^(th) framewith codebook C_(i)={c_(it); 1≦i≦2^(N)} approximated as:${S\; N\; R_{steady}} = {\frac{E_{s}}{\sigma^{2}}{\left( {N_{t} - \frac{\left( {1 - \alpha^{2}} \right)\left( {N_{t} - 1} \right)}{1 - \frac{\alpha^{2}}{\left( {1/2^{N}} \right)^{\frac{1}{N_{t} - 1}}}}} \right).}}$44. The improvement of claim 31 where channel fading is time varying andwhere the means for adjusting the codebook at each frame at steady stateconditions comprises means for generating a codebook with an averagesignal-to-noise ratio level at the t^(th) frame with codebookC_(i)={c_(it); 1≦i≦2^(N)} approximated as:C _(t) ={c _(it) =H _(ouse)(w _(t−1))[ηe ₁+√{square root over(1−η²)}f_(i)],1≦i≦2^(N)} where f_(i)=[0{tilde over (f)}_(i)]^(T), i=1, .. . , 2^(N) are N_(t)×1 column vectors, {circumflex over (f)}

[f_(i1) . . . f_(i(N) _(t) ⁻¹⁾]^(T) i=1, . . . , 2^(N) are constant(N_(t)−1)×1 column vectors with unit norm, where η is a scalar parameterand its value is given by $\begin{matrix}{{\eta = \sqrt{1 - \frac{\beta_{steady}}{\left( {1 + \sqrt{\frac{1 - \xi_{\max}}{2}}} \right)^{2}}}},} & (32)\end{matrix}$ and ξ_(max)

max_(∀i,j) Re({circumflex over (f)}_(i) ^(H){circumflex over (f)}_(j)).45. The improvement of claim 41 where channel fading is time varying andwhere the means for adjusting the codebook at each frame at steady stateconditions comprises means for simultaneously updating H_(ouse)(w_(t−1))at both the transmitter and receiver.
 46. The improvement of claim 41where channel fading is time varying and where the means for adjustingthe codebook at each frame at steady state conditions comprises meansfor simultaneously generating an identical universal codebook{circumflex over (F)} in the transmitter and receiver.
 47. Theimprovement of claim 44 where the means for generating a codebookcomprises means for generating a universal codebook satisfying thecriterion:$\hat{F} = {{\arg \; {\min\limits_{\forall\hat{F}}\xi_{\max}}} = {\arg \; {\min\limits_{\forall\hat{F}}{\max\limits_{{1 \leq i},{j \leq N}}{{{Re}\left( {{\hat{f}}_{i}^{H}{\hat{f}}_{j}} \right)}.}}}}}$48. A closed-loop encoder for transmitting a beamforming signal usingless than complete knowledge of the previous fading blocks bybeamforming a adaptive codebook of a current fading block with each timeframe comprising: a residue generator for converting an original channelvector h_(t) into a residue vector {tilde over (e)}_(t), where {tildeover (e)}_(t)=H_(ouse)({tilde over (g)}_(t))h_(t), where H_(ouse) is aHouseholder matrix; a vector quantizer coupled to the residue generatorfor converting the residue vector {tilde over (e)}_(t) to a quantizedresidue vector is denoted as {circumflex over (v)}_(t) based on aminimum distortion criterion using a distance metric d(, ), where{circumflex over (v)}_(t)=arg min_(c) _(t) _(εC) d({tilde over (e)}_(t),c_(i)), and C is a codebook, C={c₁, . . . , c_(N)} and for generating afeedback signal it from the vector quantization block; a reconstructionunit coupled to the vector quantizer for receiving the quantized residuevector {circumflex over (v)}_(t) and for generating a reconstructedchannel direction ĝ_(t) according to ĝ_(t)=H_(ouse)({tilde over(g)}_(t)){tilde over (v)}_(t); and a linear vector predictor coupled tothe reconstruction unit for generating a best estimated channeldirection {tilde over (g)}_(t) based on the past channel inputs bygenerating an output {tilde over (g)}_(t)=S(g_(t−1)) where S() denotesa predicting function, {tilde over (g)}_(t) denotes a best estimate ofthe current channel direction based on the previous channel directiong_(t−1) and for feeding back {tilde over (g)}_(t) the best estimate ofthe current channel direction based on the previous channel directiong_(t−1) to the residue generator and to the reconstruction unit.
 49. Aclosed-looped decoder for receiving a beamforming signal using less thancomplete knowledge of the previous fading blocks by beamforming aadaptive codebook of a current fading block with each time framecomprising: an inverse quantizer for receiving the feedback signal i_(t)to generate a quantized residue vector denoted as {circumflex over(v)}_(t) based on a minimum distortion criterion using a distance metricd(, ), where {circumflex over (v)}_(t)=arg min_(c) _(i) _(εC) d({tildeover (e)}_(t), c_(i)), and C is a codebook, C={c₁, . . . c_(N)}; acodebook unit coupled to the inverse quantizer for storing and providingthe inverse quantizer with a current value of the codebook C={c₁, . . ., c_(N)}; a reconstruction unit coupled to the inverse quantizer forgenerating a reconstructed channel direction ĝ_(t) according toĝ_(t)=H_(ouse)({tilde over (g)}_(t)){tilde over (v)}_(t); and a linearvector predictor coupled to the reconstruction unit coupled to thereconstruction unit for feeding back a best estimate of the currentchannel direction g, based on the previous channel direction g_(t−1) tothe reconstruction unit.